1,406 research outputs found
Numerical range for random matrices
We analyze the numerical range of high-dimensional random matrices, obtaining
limit results and corresponding quantitative estimates in the non-limit case.
For a large class of random matrices their numerical range is shown to converge
to a disc. In particular, numerical range of complex Ginibre matrix almost
surely converges to the disk of radius . Since the spectrum of
non-hermitian random matrices from the Ginibre ensemble lives asymptotically in
a neighborhood of the unit disk, it follows that the outer belt of width
containing no eigenvalues can be seen as a quantification the
non-normality of the complex Ginibre random matrix. We also show that the
numerical range of upper triangular Gaussian matrices converges to the same
disk of radius , while all eigenvalues are equal to zero and we prove
that the operator norm of such matrices converges to .Comment: 23 pages, 4 figure
Tail estimates for norms of sums of log-concave random vectors
We establish new tail estimates for order statistics and for the Euclidean
norms of projections of an isotropic log-concave random vector. More generally,
we prove tail estimates for the norms of projections of sums of independent
log-concave random vectors, and uniform versions of these in the form of tail
estimates for operator norms of matrices and their sub-matrices in the setting
of a log-concave ensemble. This is used to study a quantity that
controls uniformly the operator norm of the sub-matrices with rows and
columns of a matrix with independent isotropic log-concave random rows. We
apply our tail estimates of to the study of Restricted Isometry
Property that plays a major role in the Compressive Sensing theory
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